Solve frac{(4x^{3}y^{2})(3x^{6}y^3)}{2^4x^11y^2}

$$\frac { ( 4 x ^ { 3 } y ^ { 2 } ) ( 3 x ^ { 6 } y ^ { 3 } ) } { 2 ^ { 4 } x ^ { 11 } y ^ { 2 } }$$

Evaluate

$$\frac{3y^{3}}{4x^{2}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 6 to get 9.

$$\frac{4x^{9}y^{2}\times 3y^{3}}{2^{4}x^{11}y^{2}}$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{4x^{9}y^{5}\times 3}{2^{4}x^{11}y^{2}}$$

Cancel out y^{2}x^{9} in both numerator and denominator.

$$\frac{3\times 4y^{3}}{2^{4}x^{2}}$$

Multiply 3 and 4 to get 12.

$$\frac{12y^{3}}{2^{4}x^{2}}$$

Calculate 2 to the power of 4 and get 16.

$$\frac{12y^{3}}{16x^{2}}$$

Cancel out 4 in both numerator and denominator.

$$\frac{3y^{3}}{4x^{2}}$$

Differentiate w.r.t. x

$$-\frac{3\times \left(\frac{y}{x}\right)^{3}}{2}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12y^{5}x^{6}}{16y^{2}}x^{3-11})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3y^{3}x^{6}}{4}x^{-8})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-8\times \frac{3y^{3}x^{6}}{4}x^{-8-1}$$

Do the arithmetic.

$$\left(-6y^{3}x^{6}\right)x^{-9}$$